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Upitis, R.; Phillips, E.; Higginson, W.:
Creative Mathematics
Exploring Children’s Understanding
London: Routledge, 1997. – 185 p.
ISBN 0 415 16463 X
Nel Noddings, Stanford
Creative Mathematics is a delightful story of collaboration, mathematical curiosity, and liberal learning. The collaboration reported here is of the toughest sort – a classroom teacher, Eileen Phillips, and a university researcher,
Rena Upitis, Dean of the Faculty of Education at Queen’s
University in Kingston, Canada; it is enriched by the further collaboration of Bill Higginson, Co-ordinator of the
Mathematics, Science, and Technology Group at Queen’s.
Higginson, acting at a distance, contributed advice to the
pair working in the classroom and commentary on their
written account. Phillips and Upitis worked with third and
fourth grade children in Vancouver while Upitis was on
leave from Queen’s; Higginson encouraged, advised, and
commented.
Mathematical curiosity infected both the children and
their teachers. It is clear from the start that Phillips is a fine
teacher, but she had never before tackled the mathematics
reported here. Upitis is a music specialist and was lured
into the project in part by promises of connections between
math and music. Higginson is something of a polymath
mathematician, one interested in math and its connections
to almost everything. The children, none of whom had
listed math as their favorite subject at the beginning of
the school year, became eager for math time and sorry
when it ended. Everyone involved seems to have become
more, not less, enthusiastic as the year progressed.
The math learning was liberal in the best sense. The
activities offered set the children free to investigate matters
that excited them, and they learned a good deal more than
the usual arithmetic. Their projects took them into art,
music, technology, and invention. Along the way, their
vocabularies grew impressively.
Each author provides an introduction and commentary
in each chapter. Upitis’ many conversations with Higginson led her to explore the connection between music and
mathematics. Confessing that she had been a proficient
but not particularly interested math student, Upitis reports
being converted by the fascinating titles in Higginson’s
mathematics library. (I have seen that library and understand Upitis’ surprise and delight.) Upitis notes:
“It is not as if these books do not talk about algebra
and geometry – they do. It is that these books do not use
algebra and geometry as the endpoints in mathematics, but
rather, as tools for creation – tools that are needed to make
beautiful tessellating patterns or to understand the allure
of a snowflake” (p. 2).
With Higginson’s inspiration and the cooperation of university colleagues in Vancouver, Upitis set out to find a
teacher – collaborator.
Book Reviews
Phillips, in her introduction, expresses fears common
to classroom teachers – concern about tackling unknown
subject matter, uneasiness about working with a university
researcher, and worry over an already busy teaching life.
Besides, she was a successful teacher by all the usual standards. Why, then, should she complicate her professional
life by plunging into an alien project? However, she was
inspired and reassured by Upitis and, further, she reports
that something had been bothering her for a while:
“I felt that I was holding back. I was focusing too much on
being accountable to people outside the classroom. I wanted to
focus on the pupils and on what they might need to know. Even
when I moved to a combination of textbook work with manipulative materials and methods from my early days, I remained
unsatisfied.” (p. 8)
These remarks struck me as especially important for
American (U.S.) teachers. With the current emphasis on
higher achievement scores, many U.S. teachers feel that
accountability is pointed in the wrong direction. Instead
of being accountable to their students, teachers are forced
to consider the demands of administrators and a public
only partly aware of the ramifications of their demands.
As I travel around my country these days, I hear teachers everywhere complaining that they have been forced to
teach in a way that kills interest and does little to further
understanding. At the same time, of course, they are urged
to teach for understanding.
Phillips exudes a wisdom rarely seen in classroom teaching. Watching the pleasure and enthusiasm of her students,
she becomes dedicated to new approaches and topics. But
in response to the concerns of parents (“When will you
teach real math?”), she wisely combines textbook exercises, work sheets, and projects. She knows how to accommodate a host of conflicting needs and interests. I found
her attitude refreshing. After all, we really do not know
how much practice is required for children to learn the
skills they need to address significant mathematical problems. Alan Schoenfeld remarked on this issue that we do
not know “the degree of fluency required to do competent
work” (1994, p. 60). Thus, Phillips was not simply accommodating concerns that might have had a political impact
on her work as a teacher, but she was also sensitive to the
actual needs of her students. Neither Phillips nor Schoenfeld subscribe to the debilitating notion that students need
to learn a mass of rote skills before tackling hard and
interesting problems, but both recognize that many skills
have to be acquired somewhere, sometime, some way, if
students are to be successful in using mathematics. Indeed,
when the children were working with a project on animation, one child commented, “It’s a good thing we know
our eight times tables” (p. 65).
Higginson, in recounting his student days in mathematics, tells a familiar story. He pleased his teachers and
out-performed his peers. How many of us could tell this
story! (And how many simply gave up because they could
not “out-perform” their peers?) It was not until graduate
school that he became thoroughly engaged in mathematics. Now, of course, it is a matter of professional concern
for him to find ways to make mathematics accessible and
interesting to a great number of students.
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Book Reviews
Thus, three thoroughly motivated people launched an
impressive project. In the following chapters, they describe
how children worked on tessellations, animation, mathematical “jewels” made from paper, kaleidoscopes, and
musical composition. They learned to look for patterns
in the world (seeing tessellations everywhere), came to a
working definition of tessellation (“no floor showing”),
made important generalizations (e.g., “all quadrilaterals
will tessellate”), used symbols to describe their patterns,
applied the basic skills they had acquired earlier, and extended their vocabularies. Picture little kids easily using
words such as tessellation, animation, trapezoid, hexagon,
quadrilateral, and thaumatrope.
Phillips and Upitis put considerable emphasis on mathematical communication. This is a familiar focus today,
but their approach is more sophisticated than most. The
children were encouraged to describe their work in symbols so that others could re-create the patterns they had
constructed, and they largely succeeded. I found this work
impressive, and it contrasts sharply with many examples
of mathematical communication found in various portfolios. The difference, and it is an important one, may be
in the time spent on these projects. The children were not
rushed, and the communication was undertaken as important in itself; it was not pro forma and, because they
were asked to invent symbolic forms of communication,
it added considerably to their mathematical knowledge.
Typically, after Phillips and Upitis describe what happened in the classroom, Higginson offers a commentary
loaded with helpful references. There is clearly a lot of
help available for teachers who want to engage in the kind
of activities described here. I was reminded, as I read these
accounts, of the optimism and good sense that accompanied the best of Open Education. Other readers, too, might
want to revisit that literature. (For a personal/historical
narrative on Open Education, see F. Hawkins, 1997.)
Interestingly, although the authors do list work by David
Hawkins in their references, they do not mention Open Education. Instead, and I think this is a strategic error, they
draw an analogy between their approach to mathematics
and the whole-language approach to reading and language
arts. This is a strategic error because whole-language has
come under devastating fire in recent years (at least in
the U.S.), and anyone who wishes to use it as an example needs to defend it. I believe that a persuasive defense
can be offered, but the best use of whole-language would
probably be laid out in terms of accommodation and eclecticism – as Phillips described her own approach earlier.
Indeed, a bit of critical educational history would have
been a welcome addition to the book.
The question of what constitutes “real” problems for students arises several times, and it is sensitively discussed.
In the preface, David Pimm agrees with my comment on
“real-world” problems – that “a problem that is ‘realworld’ in the sense that adult human beings grapple with
it may not be ‘real’ at all in the school setting” (Noddings,
1994; Pimm, xii). Both Pimm and the authors clearly locate the meaning of “real” in what is real for children
– activities that engage them and from which they learn.
Thus, the “real” presents a challenge worth meeting. It
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can be fun, but it is not merely fun. Working on real
problems induces growth. On this, the authors are demonstrably Deweyan, although they do not refer to Dewey.
Higginson’s comments on technology are eminently
worth reading. They are even-handed and sensitive. He
notes the promise of technology but also its downside. Students may bog down in trivia, pursuing this and that bit
of information without studying anything in depth. They
may feel compelled to use technology even when their
own styles suggest a different approach. On this, Higginson’s report of an interview with the computer scientist
Donald Knuth is heartening. Knuth finds e-mail counterproductive for his work. He acknowledges the usefulness
of electronic mail for many occupations, but not for this
own! Comments like these can be enormously liberating.
On the positive side, Higginson’s account of what can be
found on the Web may send people scurrying for useful
information. Consonant with his even-handed analysis and
his love of books, Higginson finishes by saying, “As we
explore the Internet in the years to come, we would do well
to remind ourselves to search the shelves of second-hand
bookshops as well” (p. 130).
The last chapter, “Children as Mathematicians,” raises
several important issues. The authors frankly discuss a few
things that didn’t work. Among the unsuccessful projects
was one on codes. I was surprised by this because I recall
codes as one of the most successful topics I used with
sixth graders many years ago. Maybe the children need
to be a bit older. Maybe the teachers did not uncover the
underlying interest that might have motivated such a unit
of study. Maybe they were not all that interested themselves. I remember that my students were interested in
secret languages, writing with disappearing ink, and that
sort of thing. From there, we began to do cryptograms and
then went on to read a Sherlock Holmes story, “The Adventure of the Dancing Men” and Poe’s “Gold Bug”, both
of which involve codes. I remember the unit as great fun,
and the children learned quite a lot about the frequencies
with which various letters appear in English and about
letter combinations.
Children differ in their interests; so do teachers. That
is an important message in the last chapter. There is no
plausible reason why every group should be interested in
codes or tessellations or animations. However, a teacher’s
enthusiasm may infect the children. At least, that enthusiasm will tend to support the teacher through the hard
work of introducing and successfully completing the kind
of rich projects described here.
My own mathematical interests tend to lie in the logical
and literary. I wondered, as I read this fascinating report,
whether I would enjoy doing tessellations with children.
Maybe. But I know that I enjoy sharing and sorting out the
logical puzzles and anomalies in “Alice in Wonderland”
with students of all ages. And I am always looking for
stories that include mathematics in any form. Not long ago
I discovered the short stories of the Japanese writer Kenji
Miyazawa and found that numbers appear in almost every
one. My favorite is the story of General Son Ba-yu. A
physician examining the General learns that his patient has
been bewitched “about ten times.” The diagnostic session
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proceeds from that disclosure:
“I’d like to ask you something, then. What does one hundred
and one hundred make?”
“One hundred and eighty.”
“And two hundred and two hundred?”
“Let’s see. Three hundred and sixty, if I’m not mistaken”
“Just one more, then. What’s ten time two?”
“Eighteen, of course.” (Miyazawa 1993, p. 25–26)
Book Reviews
Noddings, N.: Does everybody count? – In: The Journal of Mathematical Behavior 13(1994), p. 89–104
Schoenfeld, A.: What do we know about curriculum? – In: The
Journal of Mathematical Behavior 13(1994), p. 55–80
Author
Noddings, Nel, Prof. Dr., School of Education, Stanford University, Stanford, CA 94305, USA.
E-mail: noddings@stanford.edu
At this point, the doctor knows just what to do and, after
treating the General, repeats the original questions. Now
the General answers correctly, and the physician says,
“You’re cured. Something was blocked up in your head,
and you were ten percent off in everything” (p. 20). Perhaps it is stories such as these that make Japanese students
so good at mathematics!
My point in introducing my own interests in logic and
literature is to remind readers that our interests differ.
Teachers and researchers who are not enthusiastic about
one set of topics can surely find another, and successful
completion of one project will almost certainly lead to
further exploration. Teachers, like children, may build on
their current interests and move well beyond them as their
confidence grows.
I may differ with the authors on one important issue. I’m
not sure that students have to learn “to think like mathematicians.” I concede that it is the popular view today in
informed circles. But why? If we were to take the advice
of all the disciplinary specialists, our children would have
to think like scientists, artists, historians, writers, linguists,
musicians, geographers, literary critics, dramatists, editors,
and mathematicians. I’m not sure this is reasonable. It
does seem reasonable, however, to analyze these ways of
thinking for those elements that students might profitably
adapt to their own purposes. I agree whole-heartedly that
students should be able to use the mathematics they learn
but surely the uses will vary with interests and talents.
We may be deluding ourselves when we try to capture the
complexity of learning to use mathematics by reducing it
to “thinking like a mathematician.” This book provides
strong evidence that kids can use mathematics while doing work similar to that of artists, engineers, film makers,
or composers. In none of these cases, however, do the
children take on the complex modes of thinking, semipermanent attitudes, and social networks characteristic of
professionals in the field. For kids, doing real mathematics involves using mathematics on problems of interest to
kids. Fortunately, we can almost always find such problems, and their mastery may well lead to new levels of
reality. Upitis, Phillips, and Higginson have demonstrated
what can be done.
References
Hawkins, F.: Journey with children: The autobiography of a
teacher. – Niwot, Colorado: University Press of Colorado,
1997
Miyazawa, K.: Once and forever. – Tokyo: Kodansha International, 1993
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